Counterexamples to maximal regularity for operators in divergence form
S. Bechtel (TU Delft - Analysis)
Connor Mooney (University of California)
Mark Veraar (TU Delft - Analysis)
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Abstract
In this paper, we present counterexamples to maximal Lp-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal L2-regularity on H-1 under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal Lp-regularity on H-1(Rd) or L2-regularity on L2(Rd).
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